Finite Difference Pricing Methods

This question focuses on numerical pricing of derivatives using finite difference schemes applied to the Black–Scholes-type parabolic partial differential equations. The candidate is asked to describe, at a high level, how one replaces continuous derivatives in time and underlying price with discrete approximations on a grid, and how this yields an algorithm that evolves option values step by step. It then narrows to the behavior of an explicit time-marching scheme, probing intuition about how the choice of grid resolution in time versus space affects the reliability of the computed prices.

Answering it well draws on familiarity with parabolic PDEs, the heat equation transformation of Black–Scholes, and the distinction between explicit and implicit schemes. The interviewer is looking for clarity on consistency, stability, and convergence, and for recognition that stability conditions link the time increment to the spatial mesh size. Strong answers discuss the trade-offs between accuracy and computational cost, explain how error can amplify in explicit schemes, and show comfort reasoning about grid design without relying on memorized formulas.

What it tests

Finite difference methods for parabolic partial differential equations, such as the heat equation or the Black-Scholes PDE, rely on discretizing both time and space, replacing derivatives with difference quotients. The stability and convergence of these methods are governed by the relationship between the time step and the square of the space step. Specifically, for explicit schemes, the method is only stable if the time step is sufficiently small relative to the square of the space step (often called the CFL condition). This arises because explicit schemes propagate information forward using only current and past values, so if the time step is too large, errors can amplify rapidly and the numerical solution can become unstable. The underlying reason is that the explicit update formula is essentially a weighted average, and the weights must remain positive and sum to one to prevent oscillations and blow-up.